2017/05/29 by Nicat Aliyev, Aliyev, Nicat, Peter Benner +7 · 1 citation
Computer Science · Mathematics · #Matrix Theory and Algorithms #Approximation Theory and Sequence Spaces #Advanced Optimization Algorithms Research
paper · pdf · doi:10.48550/arxiv.1705.10086
We are concerned with the computation of the mathcal L_\∞-norm for\nan mathcal L_\∞-function of the form H(s) = C(s) D(s)-1 B(s),\nwhere the middle factor is the inverse of a meromorphic matrix-valued function,\nand C(s), , B(s) are meromorphic functions mapping to short-and-fat and\ntall-and-skinny matrices, respectively. For instance, transfer functions of\ndescriptor systems and delay systems fall into this family. We focus on the\ncase where the middle factor is large-scale. We propose a subspace projection\nmethod to obtain approximations of the function H where the middle factor is\nof much smaller dimension. The mathcal L_\∞-norms are computed for the\nresulting reduced functions, then the subspaces are refined by means of the\noptimal points on the imaginary axis where the mathcal L_\∞-norm of\nthe reduced function is attained. The subspace method is designed so that\ncertain Hermite interpolation properties hold between the largest singular\nvalues of the original and reduced functions. This leads to a locally\nsuperlinearly convergent algorithm with respect to the subspace dimension,\nwhich we prove and illustrate on various numerical examples.\n